Question

The $g$ value on the two planets are $9.6{\text{m/s}}^2$ and $3.2{\text{m/s}}^2$ and their radii are $7200\text{km}$ and $2400\text{km}$ respectively. The ratio of escape velocities on two planets $\frac{V_{e1}}{V_{e2}}=x$. Find the value of $x$.

Answer 1

The escape velocity from the surface of a planet of mass $M$ and radius $R$ is given by

$V_e=\sqrt{\frac{2GM}{R}}$

$\Rightarrow V_e=\sqrt{2}\sqrt{\frac{GM}{R^2}\times R}$

putting the value, $g=\frac{GM}{R^2}$

$\therefore V_e=\sqrt{2}\sqrt{gR}$

So, $V_e\propto \sqrt{gR}$

$\Rightarrow \frac{V_{e1}}{V_{e2}}=\frac{\sqrt{g_1}}{\sqrt{g_2}}\times \frac{\sqrt{R_1}}{\sqrt{R_2}}$

$\Rightarrow \frac{V_{e1}}{V_{e2}}=\sqrt{\left(\frac{9.6}{3.2}\right)}\times \sqrt{\left(\frac{7200}{2400}\right)}=\sqrt{3}\times \sqrt{3}$

$\Rightarrow V_{e1}:V_{e2}=3:1$

Accepted answers : $3,3.0,3.00$

$\begin{array}{c}\text{Why this question: To make students}\\ \text{to understand the dependence of escape}\\ \text{velocity on acceleration due to gravity}\\ \text{and radius of a planet.}\end{array}$